Lecture 4 - MIT OpenCourseWare

6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007

Lecture 4-1

Lecture 4 - Carrier generation and

recombination

February 12, 2007 Contents: 1. G&R mechanisms 2. Thermal equilibrium: principle of detailed balance 3. G&R rates in thermal equilibrium 4. G&R rates outside thermal equilibrium Reading assignment: del Alamo, Ch 3. §§3.1-3.4

Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007.

MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].


Lecture 4-2

Key questions

• What are the physical mechanisms that result in the generation and recombination of electrons and holes? • Which one of these are most relevant for Si at around tempera ture? • What are the key dependencies of the most important mecha nisms? • If there are several simultaneous but independent mechanisms for generation and recombination, how exactly does one define thermal equilibrium? • What happens to the balance between generation and recombi nation when carrier concentrations are perturbed from thermal equilibrium values?




Lecture 4-3

1. Generation and recombination mechanisms a) Band-to-band G&R, by means of: • phonons (thermal G&R) • photons (optical G&R)

Ec hυ > Eg

hυ

Ev

heat thermal � generation

heat

thermal � recombination

optical � absorption

radiative � recombination

• thermal G&R: very unlikely in Si, need too many phonons si multaneously (about 20) • optical G&R: unlikely in Si, ”indirect” bandgap material, need a phonon to conserve momentum




Lecture 4-4

b) Auger generation and recombination, involving a third carrier

Ec

Ev

hot-electron� assisted� Auger� generation

hot-electron� assisted� Auger� recombination

hot-hole� assisted� Auger� generation

hot-hole� assisted� Auger� recombination

• Auger generation: energy provided by ”hot” carrier • Auger recombination: energy given to third carrier; needs lots of carriers; important only in heavily-doped semiconductors




Lecture 4-5

c) Trap-assisted generation and recombination, relying on elec tronic states in middle of gap (”deep levels” or ”traps”) that arise from: • crystalline defects • impurities

Ec

Et

Ev

trap-assisted� thermal generation

trap-assisted� thermal recombination

Trap-assisted G/R is: • dominant in Si • engineerable: can introduce deep levels to Si to enhance it




Lecture 4-6

d) Other generation mechanisms • Impact ionization: Auger generation event triggered by electric field-heated carrier • Zener tunneling or field ionization: direct tunneling of electron from VB to CB in presence of strong electric field Ec

Ec

Ev

Ev

impact ionization

Zener tunneling

• Energetic particles, such as α-particles (bad for DRAMs) • Energetic electrons incident from outside: electron microscope characterization techniques




Lecture 4-7

2. Thermal equilibrium: principle of detailed balance Define: Gi ≡ generation rate by process i [cm−3 · s−1] Ri ≡ recombination rate by process i [cm−3 · s−1] G ≡ total generation rate [cm−3 · s−1] R ≡ total recombination rate [cm−3 · s−1]

In thermal equilibrium: Ro = ΣRoi = Go = ΣGoi Actually, detailed balance is also required: Roi = Goi

for all i

In the presence of several paths for G & R, each has to balance out in detail [Principle of Detailed Balance]. [see example in notes illustrating impossibility of TE whithout de tailed balance]




Lecture 4-8

3. G&R rates in thermal equilibrium a) Band-to-band G&R • Will not consider thermal G&R as it is negligible. • Optical G&R At finite T , semiconductor is immersed in ”bath” of blackbody radi ation ⇒ optical generation. Only a small number of bonds are broken at any one time ⇒ G depends only on T : Go,rad = grad(T ) A recombination process demands one electron and one hole ⇒ R depends of nopo : Ro,rad = rrad(T ) nopo In TE, detailed balance implies: grad = rradnopo = rradn2i




Lecture 4-9

b) Auger G&R • Involving hot electrons: The more electrons there are, the more likely it is to have hot ones capable of Auger generation: Go,eeh = geeh(T )no A recombination event demands two electrons and one hole: Ro,eeh = reehn2o po In TE, detailed balance implies: geeh = reehnopo • Involving hot holes: similar but substitute no for po and eeh by ehh above.




Lecture 4-10

c) Trap-assisted thermal G&R: Shockley-Read-Hall model

Consider a trap at Et = Ei in concentration Nt.

Trap occupation probability:

f (Et) = f (Ei) =

1 ni = −EF ni + po 1 + exp EikT

Concentration of traps occupied by an electron: nto = Ntf (Ei) = Nt

ni ni + po

Concentration of empty traps: Nt − nto = Nt − Nt

ni po = Nt ni + po ni + po

Trap occupation depends on doping: • n-type: po � ni → nto � Nt, most traps are full • p-type: po � ni → nto � Nt, most traps are empty Ec

EF

Et Ev

Ec Et EF

Ev n-type

p-type




Lecture 4-11

Four basic processes:

Ec Et Ev electron� capture

electron� emission

hole� capture

hole� emission

Rates of four subprocesses in TE: • electron capture: ro,ec = ceno(Nt − nto) • electron emission: ro,ee = eento • hole capture: ro,hc = ch po nto • hole emission: ro,he = eh(Nt − nto)




Lecture 4-12

In thermal equilibrium, detailed balance demands: ro,ec = ro,ee

ro,hc = ro,he Then, relationships that tie up capture and emission coefficients: ee = ceno

Nt − nto = ceni nto

eh = chpo

nto = ch ni Nt − nto

Capture coefficients can be calculated from first principles, but most commonly they are measured. Also define: τeo =

1 Ntce

τho =

1 Ntch

τeo and τho are characteristic of the nature of the trap and its con centration. They have units of s.




Lecture 4-13

All together, rates of communication of trap with CB and VB: 1 n2i ro,ec = ro,ee = τeo ni + po

ro,hc = ro,he =

1 nipo τho ni + po

Rates depend on trap nature and doping level.




Lecture 4-14

Simplify for n-type semiconductor: ro,ec = ro,ee �

ni τeo

ro,hc = ro,he =

po τho

If τeo not very different from τho, ro,ec = ro,ee � ro,hc = ro,he The rate at which trap communicates with CB much higher than VB.

Ec

EF

Et Ev • lots of electrons in CB and trap ⇒ ro,ec = ro,ee high • few holes in VB and trap ⇒ ro,hc = ro,he small Reverse situation for p-type semiconductor.




Lecture 4-15

4. G&R rates outside equilibrium • In thermal equilibrium: n = no p = po Goi = Roi

Go = Ro

• Outside thermal equilibrium (with carrier concentrations disturbed from thermal equilibrium values): n p Gi G no Ec Go = Ro

6= 6= 6= 6 =

no po Ri

R

n=no G=R

Ev po thermal equilibrium

Ec

Ev p=po outside thermal equilibrium




Lecture 4-16

If G = 6 R, carrier concentrations change in time. Useful to define net recombination rate, U: U = R−G Reflects imbalance between internal G&R mechanisms: • if R > G → U > 0, net recombination prevails • if R < G → U < 0, net generation prevails • if R = G → U = 0, thermal equilibrium If there are several mechanisms acting simultaneously, define: Ui = Ri − Gi and U = ΣUi What happens to the G&R rates of the various mechanisms outside thermal equilibrium?




Lecture 4-17

a) Band-to-band optical G&R no

n>no

Ec

Grad = Go,rad

Go,rad = Ro,rad

Ec Rrad > Ro,rad

Ev

Ev

po

p>po with excess carriers

thermal equilibrium

• optical generation rate unchanged since number of available bonds unchanged: Grad = grad = rradno po • optical recombination rate affected if electron and hole concen trations have changed: Rrad = rradnp

• define net recombination rate:

Urad = Rrad − Grad = rrad(np − no po ) – if np > no po, Urad > 0, net recombination prevails – if np < no po, Urad < 0, net generation prevails • note: we have assumed that grad and rrad are unchanged from equilibrium




Lecture 4-18

b) Auger G&R • Involving hot electrons: no

n>no

Ec

Geeh > Go,eeh

Go,eeh = Ro,eeh

Ec Reeh > Ro,eeh

Ev

Ev

po

p>po

with excess carriers

thermal equilibrium

Geeh = geehn Reeh = reehn2p If relationship between geeh and reeh unchanged from TE: Ueeh = Reeh − Geeh = reehn(np − no po ) • Involving hot holes, similarly: Uehh = rehhp(np − no po) • Total Auger: UAuger = (reehn + rehhp)(np − no po)




Lecture 4-19

Key conclusions

• Dominant generation/recombination mechanisms in Si: trapassisted and Auger. • In TE, G and R processes must be balanced in detail. • Auger R rate in TE is proportional to the square of the ma jority carrier concentration and is linear on the minority carrier concentration. • Trap-assisted G/R rates in TE depend on the nature of the trap, its concentration, the doping type and the doping level. • In n-type semiconductor, midgap trap communicates preferen tially with conduction band. In p-type semiconductor, midgap trap communicates preferentially with valence band.

